3.996 \(\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{3} \sqrt{2} F\left (\sin ^{-1}(x)|\frac{3}{2}\right )-\frac{1}{3} \sqrt{2} E\left (\sin ^{-1}(x)|\frac{3}{2}\right ) \]

[Out]

-(Sqrt[2]*EllipticE[ArcSin[x], 3/2])/3 + (Sqrt[2]*EllipticF[ArcSin[x], 3/2])/3

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Rubi [A]  time = 0.109574, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{1}{3} \sqrt{2} F\left (\sin ^{-1}(x)|\frac{3}{2}\right )-\frac{1}{3} \sqrt{2} E\left (\sin ^{-1}(x)|\frac{3}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]),x]

[Out]

-(Sqrt[2]*EllipticE[ArcSin[x], 3/2])/3 + (Sqrt[2]*EllipticF[ArcSin[x], 3/2])/3

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Rubi in Sympy [A]  time = 16.7976, size = 26, normalized size = 0.84 \[ - \frac{\sqrt{2} E\left (\operatorname{asin}{\left (x \right )}\middle | \frac{3}{2}\right )}{3} + \frac{\sqrt{2} F\left (\operatorname{asin}{\left (x \right )}\middle | \frac{3}{2}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(-3*x**2+2)**(1/2)/(-x**2+1)**(1/2),x)

[Out]

-sqrt(2)*elliptic_e(asin(x), 3/2)/3 + sqrt(2)*elliptic_f(asin(x), 3/2)/3

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Mathematica [A]  time = 0.051221, size = 37, normalized size = 1.19 \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{2}{3}\right )-E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{2}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]),x]

[Out]

(-EllipticE[ArcSin[Sqrt[3/2]*x], 2/3] + EllipticF[ArcSin[Sqrt[3/2]*x], 2/3])/Sqr
t[3]

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Maple [A]  time = 0.018, size = 29, normalized size = 0.9 \[{\frac{\sqrt{2}}{3} \left ({\it EllipticF} \left ( x,{\frac{\sqrt{3}\sqrt{2}}{2}} \right ) -{\it EllipticE} \left ( x,{\frac{\sqrt{3}\sqrt{2}}{2}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-3*x^2+2)^(1/2)/(-x^2+1)^(1/2),x)

[Out]

1/3*2^(1/2)*(EllipticF(x,1/2*3^(1/2)*2^(1/2))-EllipticE(x,1/2*3^(1/2)*2^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-x^{2} + 1} \sqrt{-3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(-x^2 + 1)*sqrt(-3*x^2 + 2)),x, algorithm="maxima")

[Out]

integrate(x^2/(sqrt(-x^2 + 1)*sqrt(-3*x^2 + 2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{-x^{2} + 1} \sqrt{-3 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(-x^2 + 1)*sqrt(-3*x^2 + 2)),x, algorithm="fricas")

[Out]

integral(x^2/(sqrt(-x^2 + 1)*sqrt(-3*x^2 + 2)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \sqrt{- 3 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-3*x**2+2)**(1/2)/(-x**2+1)**(1/2),x)

[Out]

Integral(x**2/(sqrt(-(x - 1)*(x + 1))*sqrt(-3*x**2 + 2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-x^{2} + 1} \sqrt{-3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(-x^2 + 1)*sqrt(-3*x^2 + 2)),x, algorithm="giac")

[Out]

integrate(x^2/(sqrt(-x^2 + 1)*sqrt(-3*x^2 + 2)), x)